image/svg+xml
Electron 1 and 2
kinetic energy
Electron 1 and 2
attraction to nucleus
Electron
–
electron repulsion
This is the problem term!
2-
electron
wavefunction
Potential energy surfaces
All computational chemistry methods simplify the potential energy surface in some way
This requires chemical knowledge about the method and the potential energy surface
Resolution
Electrons essentially behave as standing waves in 3D space
Node (Zero amplitude)
Anti-nodes (Maximum amplitude)
Increasing nodes and frequency (energy)
Planck–Einstein relation relates frequency to energy:
E = hf
More nodes = higher-energy standing wave
0 Nodes
+
+
–
1 Node
2 Nodes
+
+
–
–
0 Nodes
1 Node
2 Nodes
3 Nodes
s
p
d
f
Increasing nodes and frequency (energy)
The calculations include various properties important to much of chemistry:
• Electronic/molecular orbitals
• Energy quantisation
• Dispersion interactions
• Exchange and correlation energy
• Generally much more accurate than classical methods, but take much longer
• Usually used for calculating small systems
No such thing as an absolute energy. All energies are relative in chemistry
• In computational chemistry the energy depends on the method used
• Key point: only energy differences typically matter
ΔE = +25 kJ mol
-
1
What are systems are classical methods good for?
• Biological systems
• DNA, proteins etc. (many atoms)
• Membranes (cannot consider isolated molecules)
• Materials
• Pure liquids, ionic liquids, polymers, liquid crystals (cannot consider isolated molecules, properties must be time-averaged)
• Bulk properties
• Density, diffusion, viscosity, phase changes (arise from molecular ensembles, properties change over significant time-scales)
• All these systems typically require thousands of atoms (or more) to describe them well
• These can be considered “large” systems
• Many (most) chemical systems are not in their lowest energy states under standard conditions
• Must consider only classical interactions
• I.e. no wave-behaviour of particles
• All particles treated classically i.e. as we treat macroscopic particles
• Electrons only well described by quantum methods, so classical methods
limited to atoms as the smallest particles
• To treat atoms classically we must make some (big!) approximations
U
r
AB
small k
large k
U
θ
ABC
small k
large k
A
B
C
D
ϕ
0
30
60
90
120
150
180
210
240
270
300
330
360
U
ϕ
simulated bonds behave like classical springs
• Small k = bond easily stretched (e.g. C-O, O-H)
• Large k = bond hard to stretch (e.g. C=C, C≡C)
• Small kABC = bond easily bent, e.g Si-O-Si
• Large kABC = bond angle varies little, e.g.
Torsion potentials describe ease of bond rotation
• bonding characteristics (e.g. σ and π characteristics)
• E.g. ethane H-C-C-H is staggered ethene H-C=C-H is eclipsed
Electrostatic interactions can be described using Coulomb’s law
i
j
r
U
r
Like charges
repel
U
r
Opposite charges
attract
Enables simulation of dipole-dipole interactions and H-bonding
Describe van der Waals interactions using Lennard-Jones potential
• Repulsive at short distance (electron-electron repulsion)
• Attractive at longer distance
U
r
Classical force fields
• Data files contain parameters to define many types of molecule
• Definition of atom types underpins each force-field
• Cannot simply define parameters for each element: parameters can depend on environment
• Force field files are typically thousands of lines long
• Each parameter must be carefully defined
• Force-fields vary between being very specific and very general
• Specific force-fields tend to be very good at simulating what they are designed for, but are very poor for other systems (lack transferability)
• General force-fields tend to perform reasonably well for a wide range of systems (transferable)
• Complex systems we are typically interested in have significant kinetic energy (e.g. room temperature)
• Kinetic energy is defined by the temperature, T, of a system
• We myst consider many different configurations
• A big challenge is simulating representative configurations
• The energy of a configuration is related to its probability:
• Low energy = high probability
• High energy = low probability
Classical Methods 2
Low energy, high probability
High energy, low probability
• In chemistry we typically observe a lot of molecules
• Therefore for chemical observations, the most likely configuration is an excellent representation of what we observe
• What does this mean for us? In computational chemistry, we only need to calculate the most likely configuration
Energy
Relative population
25 %
50 %
15 %
10 %
This is a configuration,
i.e. a set of occupancies
8
half
-
chair
boat
twist
-
boat
chair
Energy
25 %
50 %
15 %
10 %
Possible ratios in solution
This is a configuration, i.e. a set
of occupancies/populations of
different energy levels
25 %
25 %
25 %
25 %
This is an alternative
configuration
A chemical equivalent could be cyclohexane:
Statistical thermodynamics enables us to relate energy levels to populations (and therefore chemical properties)
𝑃
𝐸
𝑖
∝
𝑒
−
𝐸
𝑖
𝑘𝑇
Probability of a state being populated
Energy of the state
Constant
×
temperature
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Energy level
Relative population
T
5
T
10
T
100
T
Gives quantitative prediction of most likely configuration from temperature and energy levels
Molecular dynamics
• Molecular dynamics is a method by which the (many dimensional) potential
energy landscape may be explored
• Give system certain amount of energy overall (thermostat)
• Let it do what it wants over time, i.e. let the system explore the PES
Basic method
• Randomly sample the PES
• In random sampling, high energy configurations may be as
likely as low energy ones; equivalent to having infinite temperature
• Weight different configurations based on their energy
(lower energy; larger weighting)
• Definition of a force field means that the system energy is known for any set of atomic/molecular positions.
• Aims to match “real” conditions
• Input temperature/pressure/volume
• Run simulation for finite time
• Takes advantage of the fact that each configuration determines the next
• Big instantaneous jumps in atomic/molecular positions are not possible, just as they aren’t in reality
• MD trajectory maps a path through phase space (the potential energy surface)
9
10
Region
explored
Region
unexplored
Region
unexplored
Starting
point
• Temperature defines maximum energy of a system
• Energy barriers may be too high for given temperature to cross
• Other (important) low energy configurations may not be explored
• This can be a severe limitation of molecular dynamics
≠
Time average
Statistical average
Starting
point
Ergodic hypothesis
• Essentially states that a time average is the same as a statistical average
• i.e. that an MD simulation will explore all regions that should be explored according to the potential energy surface
• This is a fundamental assumption of MD
!
Multiple starting configurations can overcome energy barrier problems
MD simulation 1
MD simulation 2
MD simulation 3
Starting point
Starting point
Starting point
Statistical average
Equilibration can also be a problem with MD:
The starting geometry may not be a good representation of the final geometry
Simulation time
Region for analysis
Equilibration
• Start with “reasonable” system configuration
• Calculate energy
• Randomly perturb system to give new configuration
• Calculate new energy
- If new energy is lower than old energy:
• accept the perturbation
- If new energy is higher than old energy
• Generate random number between 0 and 1
- If Boltzmann probability of energy change is greater than
the random number, accept the perturbation
- If Boltzmann probability of energy change is less than
the random number, reject the perturbation
• Repeat
• What do we gain? Very high energy states unlikely to be sampled
• Not wasting computer power sampling unwanted regions of the PES
Metropolis Monte Carlo - a refined approach
• In practice, perturbation step usually involves randomly translating and
rotating 1 molecule, e.g. in a liquid
MD:
• Time evolution matches experimental conditions
• Cannot readily overcome energy barriers
• Can simulate time-dependent processes (e.g. diffusion, docking)
MC:
• Random selection of states does not match experiment
• Energy barriers easily overcome (with correct perturbations)
• Cannot simulate time-dependent processes
Atomic structure
How has the current view of atomic structure come about? What lead to current quantum theory?
Experiment 1: scattering of alpha-particles by gold foil
Results:
• Most particles went straight through
• Some were deflected
• A few were reflected back
Conclusions:
• Atoms are mainly empty space
• Positive charges must be concentrated in small regions
Experiment 2: Hydrogen emission spectrum
• Apply high voltage to H2 tube and measure emitted light
• Voltage electronically excites electrons
• Light emitted as they relax back to their ground state
Results: only specific wavelengths observed
• Conclusion: electrons only take specific energies
Experiment 3: X-ray/electron diffraction
X
-
ray/electron
source
X
-ray/electron
beam
Sample
Diffracted
beam
Detector
Light
Electrons
Conclusion: particles and waves can behave in the same way
Describing atoms as small positive charges (nuclei) surrounded by wave-like negative charges
(electrons) explains these observations:
- Diffuse, wave-like electrons do not interact strongly with high-energy alpha particles
- Wave-like electrons explains quantisation
- Wave-like electrons explains diffraction
n = 3
n = 4
n = 5
n = 6
Fundamental concept of quantum mechanics is that there is a wavefunction,
Ψ, that exists for any (chemical) system (i.e. particles behave as waves)
Born-Oppenheimer approximation: the assumption that the motion of atomic nuclei
and electrons in a molecule can be separated
• Nuclear motion is much slower than electron motion
• Consider nuclei to be static; just calculate for electrons
The wavefunction of a chemical system is the wave-description of the
electrons within the system
The goal of quantum methods in computational chemistry is to find the full
wavefunction of the system, Ψ(x,t)
A wavefunction can be imagined as set of standing waves that (together)
define the system
Born interpretation: Ψ
2
= probability of finding the particle anywhere in space
Shaded region must
add up to 1
x
Function is infinite
Not acceptable as a
wavefunction
x
The probability of finding the particle at x must be meaningful
For any value of x, there must only be one value of ψ(x)
Function has two
y
values at
x
= 5
Not acceptable as a wavefunction
H
Ψ
=
EΨ
The Schrodinger equation enables determination of the energy
of a system from the wavefunction
Representing a multi-electron wavefunction
How do we determine repulsion between wave-like electrons?
The orbital approximation
Assume that a reasonable approximation for a many-electron
wavefunction is to consider each electron occupying its own “hydrogenic” orbital
1 two-
electron
wavefunction
2 one
-
electron
wavefunctions
Conceptually this is OK, but we also need to think about electron spin
Spinorbital just a fancy way of defining an orbital + electron spin, e.g.
1s
α
spinorbital
:
n = 1
l
= 0
m
l
= 0
m
s
= ½
dz
2
β
spinorbital
:
n = 3
l
= 2
m
l
= 0
m
s
=
-
½
We must compine 1-electron spinorbitals in such a way that:
- No 2 electrons of the same spin can be in the same orbital (Pauli exclusion principle)
- Electrons are indistinguishable, i.e. equally associated with each spinorbital
This combination satisifies these requirements
A mathematical shorthand way of writing this expression is as a Slater determinant:
In general for N electrons we can write this sort of expression as:
Spinorbitals
Electrons
𝑒
2
4
π
ε
0
𝑟
12
• E.g. H2 molecule: the potential energy of each 1-electron orbital (spinorbital) depends on the other
• Need to approximate e- e- repulsion term in our Hamiltonian operator so it depends only on one electron
Solving this is like trying to solve 1 equation with 2 unknowns
Douglas Hartree and Vladimir Fock replaced the e- e- repulsion term in the Schrodinger equation with such an approximation
This e- e- repulsion term accounts for potential energy
in average field created by all other electrons
Final barrier to finding a solution: J and K depend of the wavefunction. Key is to use the variational principle
The exact wavefunction will correspond to the one with the lowest ground state energy, E
0
, e.g
Poor guesses
high energy
Good guess
Low energy
E
0
gives us a measure of how good our approximation is:
lower E
0
= better approximation
The best wavefunction, ψ, is the one which gives us the lowest E
0
• Guess at orbitals
- E.g. pick arbitrary parameters to describe hydrogenic orbitals
• Construct Fock operator (J & K) from these orbitals
- This means we now have an operator and orbitals
• Solve the equation using operator and guessed orbitals
- Produces new orbitals and new (lower) energy
• Construct a new Fock operator
• Repeat!
• Keep going until the input orbitals match the output orbitals
• Solution is self-consistent: output ends up the same as input
Self-consistent field (SCF) method
HF method accounts for ca. 99% of total energy
1% is due to assumption that Slater determinant is sufficient to describe wavefunction
i.e. Electron-electron interactions (correlation) are averaged; assumes electrons move independently
• In reality because electrons interact and repel each other, their motion in atoms is correlated
- When one electron is close to the nucleus, the other tends to be far away
- When one is on one side of a molecular orbital, the other tends to be on the other side
• These effects are not considered in the HF method
• A single Slater determinant also restricts us to one orbital configuration i.e. we fully define electron occupancy (not ideal in complex cases)
• The general approach of describing the overall wavefunction of a molecule as a sum of Slater determinants is the basis of the
configuration interaction (CI) method
Configuration interaction (CI)
• A solution to the problems with the HF approach is to use multiple Slater determinants (electron configurations) in our calculation
Total wavefunction =
c
0
+
c
1
π
1
π
2
π
3
π
4
π
1
π
2
π
3
π
4
e.g.
c
0
= ,
c
1
=
π
1
π
2
π
3
π
4
½
½
E.g. for He:
In this example, c
1
will be very small;
the wavefunction is mainly 1s character
• Crucially, considering more than one configuration (Slater determinant) provides information about electron correlation
i.e. we recover some of the 1% missing in HF calculations.
• Any configuration that isn't the ground state is, by definition, an excited state.
• In these calculations we use excited state configurations to help gain information about the ground state
• Adding more configurations (determinants) gives more information
• We could consider configurations where we have moved only 1 electron (singly excited states)
- configuration interaction of singles (CIS)
• We could consider configurations where we have moved up to 2 electrons
- configuration interaction of singles and doubles (CISD)
• We could consider configurations where we have moved up to 3 electrons
- configuration interaction of singles, doubles and triples (CISDT)
• "Full CI" would correspond to all possible excitations
• Full CI enables an exact solution of the Schrödinger equation (for a given set of orbital approximations) However, it is totally impractical
in most cases due to the computational power needed
• Slight alternatives give improvements to the CI method
Multi-configurational self-consistent field method (MCSCF)
• Assume excitations from low-energy occupied orbitals and to high-energy unnocupied orbitals unlikely to be significant
One varient is “Complete active space SCF” (CASSCF):
define “active” orbitals and carry out full CI within these
Much less computationally expensive than real full CI,…but still complex
Coupled Cluster (CC): another way of combining determinants
• Essentially the same process as CI, but more computationally efficient
• CC used much more commonly due to better efficiency
• CCSD(T) (approximate treatment of triples) is a typical “gold-standard” choice
Hartree Fock Method
1s, spin α
1s, spin
β
2s, spin
α
Li atom electron configuration: 1s
2
2s
1
3 spinorbitals assumed to combine to make the multi-electron wavefunction:
• Mathematical description of this combination is the Slater determinant
• Can work out the energy (via Fock operator)
• Tweak orbital shapes to minimise energy (and get the best wavefunction)
• What are we actually calculating?
• Key aspect of HF method: each electron only sees average of all the other
electrons (because of how the Fock operator works)
sees an average of + , i.e.
sees an average of
sees an average of .
=
+
+
+
i.e.
i.e.
i.e.
We have important information missing, e.g.
How does the motion of relate specifically to the motion of ?
I.e. how are their motions correlated? The energy we calculate is missing the
energy from this type of e– e– interaction, called the correlation energy.
How can we get around this with our limited description of electron
interactions?
• Answer: take multiple ‘snapshots’ of averages and combine for more detail
Post-HF methods are like taking multiple ‘noisy’ photos and combining them
to get a clearer image.
+
+
Neither original image has more detail than the other, but the combined
picture is clearer.
Post HF methods
How do we make sure our snapshots are different? Force the electrons to move
Keep the nucleus and the number of electrons identical, but optimise a
wavefunction for multiple different orbital combinations, e.g.
1s
2
2s
1
(ground state)
1s
2
2s
0
2p
1
1s
1
2s
1
2p
1
This gives 99 % of the energy
i.e. almost all the detail
These give information about
electron correlation (the 1 %)
Remember, these orbitals are only shown offset
for clarity!
Variational methods
Perturbation theory (alternative to variational methods)
• Perturbation theory is used when a (complicated) problem differs only slightly from a (simple) problem already solved
• Møller-Plesset perturbation theory does this for solving the Schrodinger equation
- Complicated problem is solving the full Schrodinger equation
- "Simple problem" is solving the Schrodinger equation without correlation effects (i.e. the HF method)
Qualitatively, MP perturbation theory adds higher excitations to Hartree-Fock theory as a non-iterative correction
• In HF theory:
H
0
ψ
0
= E
0
ψ
0
Use Fock operator instead of
H
• Use Slater determinant to describe wavefunction
• In perturbation theory: split Hamiltonian into two parts:
H
=
H
0
+λV
•
H
0
= HF Hamiltonian (Fock operator)
• λ is an arbitrary real parameter that controls the size of the perturbation (0 < λ 1)
• V is a small perturbation to the energy
• MPn truncates at λ
n
term:
- MP1 = HF
- MP2 corrects a significant amount of the correlation energy
- MP3 not significantly better than MP2
- MP4 typically account for ca. 95% of correlation energy
• MPn is not variational (unlike HF, CI, CC)
• The energy obtained is not guaranteed to always be greater than the true energy.
order
energy
HF
MP2
MP3
MP4
(Approximate) Post-HF scaling
(N is number of orbitals)
Method
Scaling
HF
N
4
MP2
N
5
MP3, CISD, CCSD
N
6
MP4, CCSD
N
7
MP5, CISDT
N
8
MP6
N
9
MP7, CISDTQ, CCSDTQ
N
10
FCI
N
!
Basis sets
• We know what atomic orbitals should look like from exact solutions
to the Schrödinger equation from the hydrogen atom
• We need to be able to describe the AOs with some flexibility in our calculations
• MOs are not exact combinations of AOs:
• In chemistry a basis set is a group of mathematical functions used to represent the
spinorbitals that combine to form the wavefunction
• A basis set typically comprises sub-groups of functions that describe the shape of
the constituent orbitals of a molecule
Slater-type orbitals (STOs)
• Replicate hydrogeinc orbitals well (solutions to the Schrodinger equation for the H atom)
ψ
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-6
-4
-2
0
2
4
6
atom 1
atom 2
atom 1 + atom 2
ψ
• STOs are accurate, but integrals are hard to calculate (which is a crucial factor) so
calculations are slow.
• General uses: high accuracy atomic/diatomic systems
Gaussian type orbitals (GTOs)
• An alternative to STOs
• Less accurate (don't replicate hydrogenic orbitals well) but fast to calculate
1s
2p
contracted Gaussian-type orbitals (CGTOs)
• STOs give better accuracy
• GTOs give better speed
• Compromise: use combination of GTOs to approximate an STO
• Slightly less accurate than real STOs, but faster
• More accurate than single GTOs, but slower
• STO-nG: STO function approximated using linear combination of n Gaussian functions
STO-3G construction: 3 gaussians to replicate 1 Slater-type orbital:
• STO-3G = single-ζ (zeta) or minimal basis set
• Only one basis function for each orbital (but each basis function made of 3 Gaussians
with a fixed ratio) – termed a CGTO
• We could use multiple basis-functions for each orbital. If two are used, it is
termed a double-ζ basis set
• A triple-ζ basis-set describes each AO with 3 basis functions
• More basis functions -> more flexibility -> better wavefunction description ->
lower-energy (better) solution to the Schrödinger equation
• But, more choice slows calculations
Split-valence basis sets
• Single-ζ basis sets (one basis function per AO)
• A poor approximation for valence orbitals; bonding distorts valence orbitals
• A good approximation for core orbitals; not involved in bonding so exhibit little distortion
3p
x
(valence orbital)
Involved in bonding so distorted
Flexible basis functions needed
2p
x
(core orbital)
Non
-
bonding so minimal distortion
Flexible basis functions not required
H
2
S
• Split-valence basis sets:
• Core orbitals described by single basis function
• Valence orbitals described by multiple basis functions
• Makes core orbital computations fast and valence orbital computations accurate
Pople basis sets
• Most commonly-used split-valence basis sets
• Developed by (Nobel Laureate) John Pople
• Standard in Gaussian software
• n-ijG (double zeta)
• n = number of Gaussian functions for each core (non-valence) orbital
• i,j = numbers of Gaussian fucntions for each valence orbitals (2 values = double zeta)
• E.g. 3-21G
• Core orbital is a CGTO comprising 3 Gaussians
• Valence described by two orbitals: 1 CGTO (2 Gaussians) and 1 single Gaussian
• n-ijkG (triple zeta)
• E.g 6-311G
• Core orbital is CGTO comprising 6 Gaussians
• Valence described by 3 orbitals: 1 CGTO (3 Gaussians) and 2 x single Gaussians.
Example: 6-31G basis set for a C atom
• Core orbitals: 1s
• CGTO comprising 6 Gaussian functions
• Valence orbital 2s
• Described by 1 CGTO comprising 3 Gaussians and 1 single Gaussian
• Valence orbital 2p
• x, y and z each described by 1 CGTO comprising 3 Gaussians plus 1 single Gaussian
1s
2s
2p
x
2p
y
2p
z
1 basis function:
•
6 Gaussians
2 basis functions:
•
3 Gaussians
•
1 Gaussian
2 basis functions:
•
3 Gaussians
•
1 Gaussian
2 basis functions:
•
3 Gaussians
•
1 Gaussian
2 basis functions:
•
3 Gaussians
•
1 Gaussian
This gives an exact expression to enable the energy of a system to be determined from the electron density (rather than the wavefunction)
HF and DFT: key difference
• Hartree Fock method is an approximate method that we solve exactly
• Exact operator, but approximate wavefunction as Slater determinant
• DFT is an exact method that we solve approximately
• Approximate operator, but exact density
• A specific DFT functional (method) is defined by how it treats this energy term
• In practice, many functionals ignore the kinetic energy correction, and many also include empirical (from experiment) corrections that
necessarily provide a correction for this
V
XC
: what is it?
Electron exchange: quantum mechanical interaction that arises from the
indistinguishable nature of electrons and from the Pauli exclusion principle
Electron correlation: the amount by which the movement of one electron is
influenced by other electrons
𝐸𝑋𝐶 assumed to be dependent on density, ρ, at position, r
• I.e. electron density at a given point defines 𝐸𝑋𝐶 exactly
• This means that (separately) the contributions to the exchange and correlation
energies are assumed to depend on the density at any given point in space
Local Density Approximation (LDA)
LDA gives good geometries but tends to underestimate exchange energy, and overestimate correlation
Generalised Gradient Approximation (GGA)
• Provides an extension to the local density approximation
• Assumes 𝐸𝑋𝐶 is dependent on density, ρ, at position, r, as well as the gradient
of ρ (i.e. how much the density is changing at that position)
Atom 1
Atom 2
Electron density
Electron density
Identical density, but different gradients,
therefore different
𝐸
𝑋𝐶
using generalised
gradient approximation
Atom 1
Atom 2
Electron density
Electron density
Identical density, therefore same
𝐸
𝑋𝐶
using local density approximation
GGA represents a significant improvement over LDA
Accuracy approaches that of methods such as MP2 and in some cases surpasses them
A DFT method (or functional) usually comprises a choice of 1 method to define exchange, and another to define correlation
Hybrid functionals
• A common approach is to incorporate some exact exchange from HF theory
• Use other sources to calculated the remainder of EXC
• Typically the most common methods used in chemistry
Method
Error /
kj
mol
-
1
HF
649
DFT (
PBE)
87
DFT (BLYP)
41
DFT (B3LYP)
40
DFT HF comparison
DFT has same computational cost, far better results (generally)
Both approximately scale as N
4
(N= number of electrons)
Limitations
• DFT Calculations are not exact solutions of the full Schrodinger equation (exact functional unknown).
• The Hohenberg-Kohn theorems apply only to ground state energies. Excited states can be predicted, but these predictions are not as good as ground-state predictions
• DFT gives inaccurate results involving weak van der Waals attractions, which are a direct result of long range electron correlation (although corrections can be made)
Single point energy
• The simplest quantum chemical calculation
• Determine total energy for fixed set of coordinates
𝐻ѱ= Eѱ
• Decide on method: determines Hamiltonian approximation (i.e. accuracy)
• E.g. HF, post-HF, DFT
• Fixed atomic coordinates determine basis function origins
• I.e. where the basis functions (atomic orbital representations) are in space
• Use self-consistent-field (SCF) method to optimise ψ
• Minimise energy for fixed coordinates
• Relies on variational principle
-1.12
-0.92
-0.72
-0.52
-0.32
-0.12
1
2
3
4
5
Energy / a.u.
Iteration
Improving wavefunction
Lowering energy
Final
wavefunction
Initial
guess
Geometry optimisations
• Probably the most common application of quantum methods
• Determine lowest-energy atomic coordinates from initial guess
• General method:
• Perform SCF calculation to determine ψ and E for initial coordinate guesses
• Displace coordinates slightly
• Perform another SCF calculation to determine ψ and E
• Repeat, e.g. until energy change is within a threshold
• Local minima can be a big problem
• Global minimum usually desired
• No way of easily checking whether a calculation has found a local or global minimum
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
0
50
100
150
200
250
300
350
400
Relative E / kJ mol
-
1
C
-
C
-
C
-
C dihedral /
°
Butane
CH
3
-
CH
2
-
CH
2
-
CH
3
Frequency calculations
• Absence of negative frequencies indicates a minimum-energy geometry
• It does not indicate a global minimum; local minima will have only positive frequencies
• Match with experimental frequencies provides confidence in calculated structures
• 1 negative frequency indicates the structure is at a transition state (saddle-point)
Potential energy scans
• A method of studying energy changes with geometry modifications
• Vary (and fix) 1 or more geometry constraints and calculate structural energies
• Typical geometry modifcations
• Atomic distance
• Bond angle
• Dihedral angle
• Provides “slice” of potential energy surface for 1 variable
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
0
100
200
300
400
Relative E / kJ mol
-
1
C
-
C
-
C
-
C dihedral /
°
Solvated systems
Adding explicit solvent molecules often adds huge amounts of complexity
• Polarisable continuum model (PCM) commonly used
• Average effect of solvent considered, rather than explicit molecules
• Does not slow down calculations much
Solvent continuum
Molecular surface
Atomic positions
Quantum methods
Classical methods
Wave-description of electrons
Standing waves in 2D
Standing waves in 3D
The wavefunction
The Schrodinger equation
H
(Hamiltonian) is the sum of the kinetic and potential energy
Solving equation for a given system should give acceptable wavefunctions and associated energies.
"Particle in a box" solutions
A hypothetical experiment where an electron is constrained between 2 infinite energy barriers)
0
0
a
V
x
0
a
0
a
0
a
x
x
x
Ψ
(x)
n
= 1
n
= 2
n
= 3
0
a
0
a
0
a
x
x
x
|
Ψ
(
x
)|
2
Wavefunctions
Probabilities
24
E
Acceptable energies
Energies in H atom are calculated
to be quantised
Constants
• ψ depends on multiple variables; we call these variables quantum numbers
Quantum numbers can be defined as:
“the sets of numerical values which give acceptable solutions to the
Schrödinger wave equation for the hydrogen atom”
Equations complex, but pictures are informative
The Schrodinger equation cannot be solved (exactly) for more than 1 electron
H atom solutions
Variational Principle
Visualisations
Density functional theory (DFT): an alternative to the wavefunction
HF and post-HF methods assume the quantum mechanical wavefunction, ψ, contains all the information about a given system
Drawbacks:
• Depends on a lot of variables: coordinates of each electron (x,y,z) and the spin state (up, down)
• Extremely difficult to calculate exactly
• Non-intuitive: gives the energy, but does not make much sense in itself
Dependence of ψ on electron coordinates suggests that the electron density, ρ, may be a suitable alternative
First Hohenburg-Kohn theorem
• The electron density of any system determines all ground-state properties of the system
• Suggests the density must define the external potential of the system, therefore the Hamiltonian, 𝐻, the wavefunction, ψ, and the energy of a system
- Just as the wavefunction, ψ, defines all properties of a system, including the electron density, ψ
2
.
• This theorem does not give any information as to how the density defines properties!
Second Hohenburg-Kohn theorem
• The correct ground state density for a system is the one that minimizes the total energy through the functional E[n(x,y,z)]
• I.e. the variational principle applies
• It is desirable to have an equivalent of the Hamiltonian in the Schrodinger equation that acts on the electron density to give an energy
Starting point: assume that electrons do not interact non-classically
Kinetic energy
of non
-
interacting
electrons
Nuclear
-
electron
potential energy
Electron
-
electron
potential energy
𝐾𝐸
𝑛𝑖
[𝜌 (𝒓)] + 𝑉
𝑛𝑒
[𝜌(𝒓)] + 𝑉
𝑒𝑒
[𝜌 (𝒓)]
This provides an exact Hamiltonian for a system of electrons that only interact classically. Can solve it just like we do for the Schrodinger equation
We still have a problem: we know electrons interact non-classically, and we know these interactions are significant
• Solution = add corrections to our energy terms:
𝐾𝐸
𝑛𝑖
[𝜌 (𝒓)] + 𝑉
𝑛𝑒
[𝜌(𝒓)] + 𝑉
𝑒𝑒
[𝜌 (𝒓)] + Δ𝐾𝐸
𝑛𝑖
[𝜌 (𝒓)] + Δ𝑉
𝑒𝑒
[𝜌 (𝒓)]
Kinetic energy
of non
-
interacting
electrons
Nuclear
-
electron
potential energy
Electron
-
electron
potential energy
Correction to electron
-
electron potential energy
due to interaction
Correction to kinetic
energy of electrons
due to interaction
The correction is often abbreviated as 𝑉
𝑋𝐶
or
E
𝑋𝐶
This energy term combines all non-classical (quantum) interactions
Force fields
Through-bond interactions
Through-space interactions
Bond stretching
Angle bending
Dihedral torsion
Electrostatic interactions
Lennard Jones interactions
Sampling representitive configurations
Monte Carlo
MD MC comparison
Calculation types
Key concepts
Anything collection of objects and positions may be described by a potential energy surface
In chemistry we consider collections of electrons, nuclei and atoms.
The energy of a chemical system varies with the location of all the constituent particles
Full PES = energy known for every possible combination of variables
Dimensions of PES scale with number of particles as 3N - 6
(Translation and rotation of entire system not relevant to energy)
Typically impossible to calculate full PES: too many combinations of variables
Simplifying the PES
Omit some areas
Calculate route through PES
Molecular energies
Energies vs. populations (statistical thermodynamics)
How do we use statistical thermodynamics?
Some calculations do not need to use this relationship, e.g. in molecular dynamics the energies and populations are both calculated
Some calculations automatically include statistical thermodynamics, e.g. Metropolis Monte Carlo methods
Some calculations must have this releationship applied subsequently, e.g. if we want to know relative populations from a quantum method
(which only gives us a system energy)
AP0627 Theoretical and
Computational Methods
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AP0627 course overview